Moons are, we think, very common. After all, every single planet that humans have lived on has had at least one moon.
Most of the planets in our solar system have at least one moon, and some large planets have lots.
By extension, why shouldn't we expect that exoplanets would also have moons?
But how do we find them?
There are a few methods. I'll try to explain the "transit" method for exomoon detection here, although there are others.
Scientists record the amount of light coming from a star. A dip in the luminosity may indicate the transit of an exoplanet.
By extension, higher-resolution measurements may be able to detect the transit of an exoplanet with a moon, but it would depend on the arrangement of star, planet and moon to be just right.
Have a look at the sequence in the graphic above, and my scribbled light trace over time. In the first panel, the planet and moon are approaching transit. Here the luminosity would still be 100% of normal.
In the second panel, only the moon is transiting, and so the light from the star will have fallen by a tiny amount, say, to 99%. It's these smaller dips moon-seekers are looking for. When the planet joins the moon it transit, the light received from the star dips more significantly, say, to 95% of normal. This is the third panel. In the fourth panel, I've represented the moon leaving transit first, although that's not always going to be the case. Luminosity rises slightly, to say, 96%. The last panel is back to normal luminosity, with the transit finished.
This is, of course, an idealised demonstration, and scientists have to contend with all manner of complications.
If exomoons are as numerous as we think, the probability of finding them in "Goldilocks" zones is worth considering. If life (as we know it, Jim) is out there, it may be on a rocky moon, possibly with an atmosphere, orbiting a gas giant.
This was based partly on Shannon Hall, "the Hunt for the First Exomoons", Sky and Telescope (US Edition), September 2020